arXiv:1406.4499v1 [hep-th] 17 Jun 2014 Off-shell structure of twisted (2,0) theory Ulf Gran, Hampus Linander, Bengt E.W. Nilsson Department of Fundamental Physics Chalmers University of Technology S-412 96 Göteborg, Sweden ulf.gran@chalmers.se, linander@chalmers.se, tfebn@chalmers.se Abstract AQ-exact off-shell action is constructed for twisted abelian (2,0) theory on a Lorentzian six-manifold of the formM1,5=C×M4, whereCis a flat two-manifold andM4is a general Euclidean four-manifold. The properties of this formulation, which is obtained by introducing two auxiliary fields, can be summarised by a commutative diagram where the Lagrangian and its stress-tensor arise from theQ-variation of two fermionic quantitiesVand λµν. This completes and extends the analysis in [1]. 1
Contents 1Introduction2 2The twisted theory3 3Conserved stress-tensor5 4Q-exact action7 4.1E-sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 4.2F-sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 5Conclusions12 AMetric variation of self-dual forms13 1Introduction In this note we consider twisted (2,0) theory on a Lorentzian six-manifold of the formM1,5=C×M4whereCis a flat, Lorentzian two-manifold [1]1. This setup is interesting since it could give some insight into the conjectured correspondence between four-dimensional gauge theory and two-dimensional CFT known as the AGT-correspondence [2, 3]. This is one part of a larger web [4–8] of dualities and relations that can be derived assuming the existence of the elusive superconformal theory in six dimensions known as (2,0) theory [9, 10]. In previous work [1] the twisted theory onM4was calculated explicitly in terms of the free tensor multiplet. It was shown that on a flat background there is aQ- exact and conserved stress tensor but that these properties did not immediately extend to a general curvedM4. The problem was located to the stress tensor for the bosonic self-dual two-form which turned out not to be conserved on a general four-manifold.However, this issue needs to be remedied since the procedure of topological twisting should result in theories withQ-exact stress tensors defined on a generally curved background [11–13]. Here we construct an action for the full theory that isQ-exact off-shell using two different kinds of auxiliary field.The free theory splits into two parts, one of which is equivalent to Donaldsson-Witten theory, and hence this sector can be taken off-shell following the standard techniques described in [14–16]. 1The twisting is carried out in Minkowski signature where the non-compact part of the Lorentz group prevents a full twist but where a low energy limit still produces supercharges with the required properties for a topological theory. 2
In the other sector there is a self-dual tensor field whose presence in theQ transformation rules leads to an unwanted metric dependence.However, by the introduction of an auxiliary vector field we are able to eliminate this metric de- pendence in a similar fashion as in [17].Also in this sector, this step leads to a formulation where the scalar supercharge is nilpotent off-shell and after construct- ing aQ-closed and covariant Lagrangian for the entire theory these properties become manifest also for the stress-tensor. For the bosonic self-dual two-form we have also added certain curvature terms to its equation of motion, which becomes possible after the twisting is performed. Note that these terms cannot be obtained in the original (2,0) theory in six dimensions but they are known from the closely related interacting theory constructed in five dimensions in [5]. With these modifications of the theory that is naively obtained from twisting the six-dimensional (2,0) theory onM4, we find an off-shell theory whose metric variations andQtransformations commute.This feature then implies that the stress tensor can be derived from a fermionic quantityV(given below) either by going via the Lagrangian or viaλµν(whereTµν={Q,λµν}).In section 4 this is summarized in acommuting squarewhose corners represent the four quantities involved, i.e.,Tµν,λµν,Vand itsQtransform, the Lagrangian. In section 2 we review the four-dimensional theory obtained by twisting the six-dimensional (2,0) theory onC×M4. The problem encountered previously and its resolution are briefly explained in section 3. In section 4 we construct an off- shell formulation including aQ-exact action. Finally, in section 5 we summarise and comment on the results. 2The twisted theory For the convenience of the reader we here give a short review of the twisted the- ory, for details see [1]. On a general background the six-dimensional (2,0) theory admits no twist that preserves any supersymmetry since the Spin(5) R-symmetry cannot be used to fully twist the supercharges transforming in the larger six di- mensional Lorentz group Spin(1,5). However, on specific backgrounds such as the one considered here of the formM6=C×M4, the Lorentz group is small enough. Here we twist by considering a new SU(2)′as the diagonal embedding SU(2)′= SU(2)r×SU(2)R,(1) where the six dimensional Lorentz group is Spin(1,1)×SU(2)l×SU(2)rand the R-symmetry subgroup is given by SU(2)R×U(1)R∼= Spin(3)×Spin(2)⊂Spin(5)R.(2) 3
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